Weak money is good for exporters and bad for importers, strong money is bad for exporters and good for importers. So companies like Sony and Nintendo make money when the yen is weak, and lose money when the yen is strong. For example, say Sony sells $1 billion worth of shit in the US: if the yen is weak, say 100 yen per dollar, that $1 billion translates into 100 billion yen, but if the yen is strong, say 75 yen per dollar, it only translates into 75 billion yen. Because their business is based in Japan and based on the yen they primarily care about how much yen they make, so they want the yen to be weaker. Importers are the opposite, because if you're buying stuff from outside of Japan a strong yen means you can buy more things per yen, so they're effectively cheaper for you.
The reason the $=£=€ thing is bullshit is because it has nothing to do with the strength of the economies. The base value of the currency is completely arbitrary, the only thing the economy affects is changes in the relative values. Like say Scotland secedes from the UK and starts making their own currency: they could say that the base unit of their currency, 1 Scot, is worth $10 right now. That doesn't mean their economy is good, it just means that's the value they picked. After that starting point the value of the Scot will fluctuate based on the relative strengths of the economies. So now say their economy crashes and they have rampant inflation, and now 1 Scot is only worth $3: it's still worth a hell of a lot more than $1, even though their economy is down the toilet, so setting digital prices at $1 = 1 Scot would be completely ridiculous. The same is true with the £ and €: each is worth more than the dollar but it has nothing to do with the relative strengths of the economies.